Cossacks were less traditionally 'Ukrainian' and more their own culture and tribes which often (but not always, see, e.g. the Don Cossacks) spoke Ukrainian. Cossacks were seminomadic raiding peoples from the northern areas around the black sea, as distinct from the settled Slavic communities like Kiev.

The name Cossack likely comes from the Turkish 'Kazak' meaning 'free man' or an adventurer. They are first mentioned by the Genoese who controlled the area of Crimea in the 14th century and that reference is likely to raiders who were likely some mixture of Tatar and Cuman horsemen who'd been pushed west and dislocated by Mongol conquest.

Initially there were both town and 'free' cossacks, with the former acting more as militias and armed protectors while the latter were semi-nomadic raiders. The cossacks we think of these days tend to be these latter, 'free' cossacks.

Their organization was largely in opposition to settled rule, and ranks would swell with previously-settled townspeople who'd make their way down to escape rule they disliked. Free cossacks eventually ended up as a major force to be reckoned with for both the Crimean Khanate tatars and the Russian empire to the north.

Their integration into the Russian empire (rather than status as a pest on its outskirts) began in earnest in 1570 when Ivan the Terrible recognized them, supported them in their war against the tatars, and they supported him in some of Russia's foreign adventures. These Cossacks formed the Don Host (from being based on and around the Don river and settling a stronghold in Cherkassiya) and, after these wars, formed a sort of buffer-region between the Russian empire, the Turks to the south and the tatars, cumans, and other steppe peoples to the east.

They were used as autonomous border troops and crack cavalry for the Russian empire, and particularly its expansion east and south. Likewise, their autonomy was recognized and they were allowed to largely self-govern in exchange for repressing the enemies of the Russian tsar (something which continues to this day, to an extent, as when Putin used a Cossack host to whip and repress protestors during the Sochi Olympics).

There's a whole lot more to them as a peoples, including really interesting cultural quirks which come out of their culture being based in opposition to that of settled people who Cossacks might traditionally consider 'unfree'. For example, the fact that while land in a Cossack host was, initially, held in common and available for any member to use, one prohibited early use of land was for farming. Because Cossacks were in opposition to something that requires you to be so settled.

But, in any case, none of that interesting stuff is necessary to answer your question. As you can likely see at this point, though Cossacks originated and were often associated with land which is now in Southern Ukraine, they were not culturally Ukrainian. Not only did they have their very own culture, they often ranged well outside the territorial boundaries of Ukraine and the largest and most recognized host was settled in what has for a long time been considered to be Russia (the northern Caucuses and the northeastern areas of the black sea) so it wouldn't be fair to consider them Ukrainian either from a cultural, geographical, or ethnic perspective although they were 'adjacent' in some senses.

The Cossacks are, thus, not Ukrainian. For many centuries they enjoyed special privileges and semi-autonomy to a level rarely seen in Russia which was granted by the Tsar and the Tsar's support kept them from being overwhelmed by the numerically superior Turks and Tatars who they raided. Neither they nor the Russian rulers saw Cossacks as Ukrainians, they were seen as Cossacks. As such, the repression of Ukraine under the Soviets had little to do with repressing them (though they might have participated in doing some of the repressing), it was repressing a whole different peoples who the Cossacks happened to live near and, occasionally, speak a similar language.

For a lot more background on Cossacks, check out Albert Seaton's The Cossacks, 1972, ISBN 9780850451160